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On convergence determining and separating classes of functions. (English) Zbl 1210.60011

Let \(E\) be a topological space. In the theory of probability measures, two problems are very important:
1) To define a determining class of functions, i.e., a class \({\mathcal M}_1\) of functions such that the equality \[ \int_Ef\,dP=\int_Ef\,dQ\quad {\text{for all}}\quad f\in{\mathcal M}, \] implies the equality \(P=Q\) for Borel probability measures on \(E\);
2) To define a convergence determining class, i.e., a class \({\mathcal M}_2\) of functions such the relation \[ \int_Ef\,dP_n=\int_Ef\,dP\quad {\text{for all}}\quad f\in{\mathcal M}, \] implies the weak convergence \(P_n\) to \(P\) as \(n\to\infty\).
Of course, the classes \({\mathcal{M}}_1\) and \({\mathcal{M}}_2\) must be as narrow as possible.
Let \(M(E)\) denote the set of Borel measurable functions. Then the set \({\mathcal M}\subset M(E)\) separates points if, for \(x\neq y\in E\), there exists a \(g\in{\mathcal M}\) such that \(g(x)\neq g(y)\), and \(\mathcal M\) strongly separates points if, for every \(x\in E\) and a neighborhood \(O_x\), there is a finite collection \(\{g^1, \dots, g^k\}\subset {\mathcal M}\) such that \[ \inf_{y\notin O_x}\max_{1\leq l\leq k}\left|g^l(y)-g^l(x)\right|>0. \]
S. N. Ethier and T. G. Kurtz [Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics. New York etc.: John Wiley & Sons (1986; Zbl 0592.60049)] proved that if \(\mathcal M\) separates points and is an algebra of continuous function, then \(\mathcal M\) is a determining class on Polish spaces. Similarly, they showed that if \(\mathcal M\) strongly separates points and also is an algebra of continuous bounded functions, then \(\mathcal M\) is a convergence determining class on Polish spaces.
The authors apply homeomorphism methods, and generalize and simplify the above results for more general spaces including incomplete metric spaces, Skorokhod spaces, and what is very important, Lusin spaces.

MSC:

60B10 Convergence of probability measures
60B05 Probability measures on topological spaces

Citations:

Zbl 0592.60049
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References:

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